我在高中和大学时讨厌透了基于极限的微积分。后来我读了《初等微积分：无穷小分析法》（Elementary Calculus: An Infinitesimal Approach，http://www.math.wisc.edu/~keisler/calc.html），结果只用了几页的篇幅，一切就全讲明白了。真让人火大，大多数数学课程就是不肯抛弃那些老旧、臃肿、过度……
I loathed limit-based calculus in High School and College. Later I read Elementary Calculus: An Infinitesimal Approach http://www.math.wisc.edu/~keisler/calc.html and it all came clear in a fraction of the pages. It's infuriating that most math curricula won't drop those old, bloated, overly…
……翻看《微积分简易读本》（Calculus Made Easy）时，我发现很有意思的一点：书一开头讲的是“关于不同程度的小”，极限只是定性提了一下，并没有正式展开。这让我想起了《初等微积分》[1] [2]的前言，那是一本极好的教材，是我在用极限学完微积分很多年后才发现的：> 微积分最初是借助直观的无穷小概念（或者说是无穷……）发展起来的。
…through Calculus Made Easy, I found it curious that the text begins with “On Different Degrees of Smallness” and the limit is mentioned qualitatively but not used formally. It reminds me of the preface to Elementary Calculus [1] [2] an excellent text that I discovered years after learning the subject using the limit: > The calculus was originally developed using the intuitive concept of an infinitesimal, or an infinitely…
I don't have a complete answer for you, but I linked to this book a few days ago. It's pretty good. http://www.math.wisc.edu/~keisler/calc.html Elementary Calculus: An Infinitesimal Approach for a mathematically rigorous course in infinitesimal calculus. I think it is much more intuitive than typical limit calculus.
Another recommendation - Keisler's Elementary Calculus: An Infinitesimal Approach http://www.math.wisc.edu/~keisler/calc.html (pulled from this old slashdot thread) http://books.slashdot.org/story/04/03/04/028253/Five-Free-Ca...
……在机器上实现的无穷小（也称为“非标准分析”）。如果你喜欢这种方法，可以通过阅读 H. Jerome Keisler 的《初等微积分》进行更通用、更严谨的探索，第二版在网上可以免费获取：http://www.math.wisc.edu/~keisler/calc.html。现在第三版已经出纸质书了。我断断续续用它学微积分有一阵子了……
…with infinitesimals (also called "nonstandard analysis") implemented on the machine. If you like the approach, a more general and rigorous investigation can be had by reading H. Jerome Keisler's book Elementary Calculus, which is freely available online in the 2nd edition here: http://www.math.wisc.edu/~keisler/calc.html The third edition is now in print. I've been studying calculus with it off-and-on for a while…
Other Calculus book worth mention is "Elementary Calculus: An Infinitesimal Approach", by Jerome Keisler. One thing that makes learning Calculus harder than necessary is the clumsy epsilon and delta formalism, that became sort of a COBOL for Calculus. This is…
我在高中和大学时讨厌透了基于极限的微积分。后来我读了《初等微积分：无穷小分析法》（Elementary Calculus: An Infinitesimal Approach，http://www.math.wisc.edu/~keisler/calc.html），结果只用了几页的篇幅，一切就全讲明白了。真让人火大，大多数数学课程就是不肯抛弃那些老旧、臃肿、过度……
I loathed limit-based calculus in High School and College. Later I read Elementary Calculus: An Infinitesimal Approach http://www.math.wisc.edu/~keisler/calc.html and it all came clear in a fraction of the pages. It's infuriating that most math curricula won't drop those old, bloated, overly…
……翻看《微积分简易读本》（Calculus Made Easy）时，我发现很有意思的一点：书一开头讲的是“关于不同程度的小”，极限只是定性提了一下，并没有正式展开。这让我想起了《初等微积分》[1] [2]的前言，那是一本极好的教材，是我在用极限学完微积分很多年后才发现的：> 微积分最初是借助直观的无穷小概念（或者说是无穷……）发展起来的。
…through Calculus Made Easy, I found it curious that the text begins with “On Different Degrees of Smallness” and the limit is mentioned qualitatively but not used formally. It reminds me of the preface to Elementary Calculus [1] [2] an excellent text that I discovered years after learning the subject using the limit: > The calculus was originally developed using the intuitive concept of an infinitesimal, or an infinitely…
我没有一个完整的答案，但我几天前分享过这本书的链接。写得很棒。http://www.math.wisc.edu/~keisler/calc.html 《初等微积分：无穷小分析法》，适合想严谨学习无穷小微积分课程的人。我认为它比常见的极限微积分要直观得多。
I don't have a complete answer for you, but I linked to this book a few days ago. It's pretty good. http://www.math.wisc.edu/~keisler/calc.html Elementary Calculus: An Infinitesimal Approach for a mathematically rigorous course in infinitesimal calculus. I think it is much more intuitive than typical limit calculus.
再推荐一本——凯斯勒（Keisler）的《初等微积分：无穷小分析法》http://www.math.wisc.edu/~keisler/calc.html （从这个老 Slashdot 帖子里翻出来的）http://books.slashdot.org/story/04/03/04/028253/Five-Free-Ca...
Another recommendation - Keisler's Elementary Calculus: An Infinitesimal Approach http://www.math.wisc.edu/~keisler/calc.html (pulled from this old slashdot thread) http://books.slashdot.org/story/04/03/04/028253/Five-Free-Ca...
……在机器上实现的无穷小（也称为“非标准分析”）。如果你喜欢这种方法，可以通过阅读 H. Jerome Keisler 的《初等微积分》进行更通用、更严谨的探索，第二版在网上可以免费获取：http://www.math.wisc.edu/~keisler/calc.html。现在第三版已经出纸质书了。我断断续续用它学微积分有一阵子了……
…with infinitesimals (also called "nonstandard analysis") implemented on the machine. If you like the approach, a more general and rigorous investigation can be had by reading H. Jerome Keisler's book Elementary Calculus, which is freely available online in the 2nd edition here: http://www.math.wisc.edu/~keisler/calc.html The third edition is now in print. I've been studying calculus with it off-and-on for a while…
另一本值得一提的微积分书是 Jerome Keisler 写的《初等微积分：无穷小分析法》。让微积分学习变得不必要地困难的元凶之一，就是笨拙的 epsilon-delta 形式化表述，它简直成了微积分界的 COBOL。这种……
Other Calculus book worth mention is "Elementary Calculus: An Infinitesimal Approach", by Jerome Keisler. One thing that makes learning Calculus harder than necessary is the clumsy epsilon and delta formalism, that became sort of a COBOL for Calculus. This is…